Beam tomography as described for example in U.S. Pat. No. 9,013,956 (“the '956 Patent”) improves estimations of seismic velocities by finding velocity corrections that improve the alignment of beams with a seismic image. The beam tomography method described in the '956 Patent includes a step of determining velocity model corrections that improve a match between beams formed by locally steering recorded data and beams formed by modeling the recorded beams using the current velocity model and current image. As such, velocity model updates can be obtained by minimizing the difference between recorded data and synthetic data formed by forward modeling computations.
The '956 Patent describes a so-called maximum-correlation shift method (MCS), which includes the step of cross-correlating data and modeled beams to determine the quantitative match between pairs of data and modeled beam. Each of the beam pairs corresponds to a single ray path through the earth. The complete ray path has a segment beginning at a seismic source and another segment terminating at a receiver. The beams are functions of travel time along their associated ray path. The two beams in a pair are cross-correlated over a range of travel time shifts. The time shift that maximizes the cross-correlation of each beam pair is captured and used as a data value for the tomography. This matching by cross correlation is done for many raypaths, typically on the order of 107 for the current applications.
Next, the velocities along each raypath must be changed to fit the time shift measured between the data and modeled beam. Each raypath and corresponding time shift generates an equation that linearly relates the velocities changes along the raypath to the measured travel time shift. For MCS implementations of beam tomography, this large, sparse system of equations is solved by an iteratively reweighted least squares inversion of a large, sparse matrix. The input data for the MCS method are alignment shifts with a single shift value represents the proper alignment of each pair of modeled and data beams. The success of the MCS implementation depends on the cross correlations providing good measurements of the misalignments between beam pairs. Unfortunately these measurements can be degraded by cycle skips, which cause the cross-correlations to misrepresent the correct alignments.
To some degree, cycle skips can be tolerated by driving the reweighted least-squares solution toward an L1 norm, which better handles the data outliers caused by cycle skips. This reweighting process, however, can only handle a small degree of cycle skipping, which means that the data must have low levels of noise.
An example of the cycle skipping problem is shown in FIGS. 1A and 1B. FIGS. 1A and 1B shows pairs of data and modeled beams: the dark traces are beams computed by localized slant stacking of the recorded data, and light traces are computed by modeling computations using the an earth model to describe the subsurface velocity structure and a seismic image to describe the subsurface reflectors. The light and dark traces are shown in pairs, such that each pair corresponds to a ray path through the earth. FIG. 1A shows the unaligned traces, and FIG. 1B shows the traces after alignment. Note, the horizontal lines through the middle of FIGS. 1A and 1B are timing lines corresponding to ray trace arrival times.
With reference to FIG. 1A, the two traces in each pair have some amount of misalignment, which is caused by errors in the earth model and image. The proper alignment of most traces is unambiguous and can easily be measured by finding the cross-correlation maximum. In FIG. 1B, the traces are aligned by shifting them to obtain maximum cross correlations—between light and dark traces. The line at the top of FIG. 1B (“Shifts”) shows the time shifts needed to maximize the cross correlations. Note, however, the mid offsets near the middle of FIG. 1B. Here, rapid variations in time shifts are used to align the traces. These rapid variations are caused by cycle skips, for which the maximum cross-correlation alignment differs from the correct alignment by one or more cycles of the waveform. As such, it is not clear from the data alone which of the local maximum in the cross-correlation function corresponds to the correct alignment. The global maximum may not be correct due to noise or numerical approximation.
Additionally, in cases where the data contain strong coherent noise caused by multiple reflections, such as in FIG. 2A, the traces can be shifted to form an apparently well aligned event as shown near the center of FIG. 2B. However, it might be that the cross-correlation maximum does not correspond to the proper trace alignment. Although there appears to be convincing alignment of the events near the middle of FIG. 2B, the extreme irregularity of the time shifts plotted at the top of this panel suggests the presence of a spurious alignment caused by the coherent noise.
As such, a problem with the previously disclosed MCS method is that the step of measuring the time shifts between beams of a beam pair is independent from the step of inverting for the velocity corrections in the earth model. Alignments performed solely by finding maximum cross-correlations are prone to cycle skips because there is no discrimination based on whether a shift corresponds to a reasonable change in velocity and whether this shift is in accord with the shifts measured between other beams pairs. Measured shifts are fixed and are subsequently used in a linear inversion for velocity. This linear inversion has no provision for correcting outlier measurements that are caused by cycle skips or for discarding shifts that are spurious alignments with coherent noise.
As such, a need exists for a travel-time reflection tomography method that is less susceptible to the effects of cycle skipping and a spurious alignment with coherent noise.